JOURNAL OF COMPUTATIONAL BIOLOGY Volume 19, Number 3, 2012 Review Article # Mary Ann Liebert, Inc. Pp. 316–336 DOI: 10.1089/cmb.2011.0234 Probabilistic Logic Methods and Some Applications to Biology and Medicine NIKITA A. SAKHANENKO1 and DAVID J. GALAS1,2 ABSTRACT For the computational analysis of biological problems—analyzing data, inferring networks and complex models, and estimating model parameters—it is common to use a range of methods based on probabilistic logic constructions, sometimes collectively called machine learning methods. Probabilistic modeling methods such as Bayesian Networks (BN) fall into this class, as do Hierarchical Bayesian Networks (HBN), Probabilistic Boolean Networks (PBN), Hidden Markov Models (HMM), and Markov Logic Networks (MLN). In this re- view, we describe the most general of these (MLN), and show how the above-mentioned methods are related to MLN and one another by the imposition of constraints and re- strictions. This approach allows us to illustrate a broad landscape of constructions and methods, and describe some of the attendant strengths, weaknesses, and constraints of many of these methods. We then provide some examples of their applications to problems in biology and medicine, with an emphasis on genetics. The key concepts needed to picture this landscape of methods are the ideas of probabilistic graphical models, the structures of the graphs, and the scope of the logical language repertoire used (from First-Order Logic [FOL] to Boolean logic.) These concepts are interlinked and together define the nature of each of the probabilistic logic methods. Finally, we discuss the initial applications of MLN to ge- netics, show the relationship to less general methods like BN, and then mention several examples where such methods could be effective in new applications to specific biological and medical problems. Key words: Bayesian networks, first-order logic, hierarchical Bayesian networks, machine learning, Markov logic networks, probabilistic Boolean networks, probabilistic graphical models, propositional logic. 1. INTRODUCTION ogic and probabilistic graphical models are natural tools of computing and have been studied Land used in computer science for many years. Some of these methods have emerged in recent years as promising approaches to a range of problems in artificial intelligence. In biology and medicine, where many of the computational problems involve data analysis and the inference of underlying complex models, simple forms of these methods, such as Hidden Markov Models (HMM), and simple forms of Bayesian Networks 1The Institute for Systems Biology, Seattle, Washington. 2Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Esch-sur-Alzette, Luxembourg. 316 PROBABILISTIC LOGIC METHODS IN BIOLOGY 317 (BN), have become more commonly used in recent years (Baldi and Brunak, 2001). There is now a wide range of these methods that have been usefully applied to biological and medical problems, but it is often difficult to see in the reports of successful applications in the literature how they are related to one another. In this review, we attempt to address this difficulty and elucidate some of the key relationships, not specifically for the experts in the field, but more for the computational biologists that are grappling with the realities and complexities of current problems in systems biology, genetics and their applica- tions to modern medicine. This review is not a comprehensive survey of methods, nor of the specific techniques and implementations in the literature, but rather is focused on providing a map of some of the key concepts that we hope will allow computational biologists to see the relationships, contrasts, and overlaps among widely used methods. Even more importantly, in our view, we hope to point out where new applications of probabilistic logic methods may provide powerful new approaches to the sometimes dauntingly complex problems of understanding biological systems. Our plan for this review is to begin with the most general combination of logic and probability, and illustrate how simplifying restrictions and constraints lead to more familiar methods, in order to elucidate the key relationships, and to indicate where some power is lost and where efficiency is gained. There are several relevant logical systems and several different forms, and representations, of probability distributions. These are briefly summarized below. First-Order Logic (FOL) is a very general formal logic, and represents a more powerful tool for ex- pression of logical relationship than propositional logic (Ershov and Palyutin, 1986). Using propositional logic, we are not able to express assertions about classes of objects or events. FOL is considerably richer than propositional logic and allows just these expressions. FOL allows the propositional symbols to have arguments that range over elements of sets, which allows us to make assertions about the sets or classes. FOL can also deal with recursive statements, while propositional logic cannot. Probability distributions can involve a wide range of simple and complex dependencies and are often represented in graphical form in statistics, where the independencies of the variables represented as nodes are encoded in the edges. In this review, we illustrate the key relationships by beginning with the most general and proceeding to more restrictive, and constrained representations. While there are some sig- nificant differences in the graphical representations used in various methods the logical components of the probabilistic logic are where most of the restrictions occur. Markov Logic Networks (MLN) represent a new and general approach to modeling based on full FOL and Markov Random Fields (MRF) (Richardson and Domingos, 2006). In addition to high representational power of FOL, MRFs provide a very compact way of representing probability distributions and are very useful for modeling and reasoning in noisy, uncertain environments. For physicists, MRFs are probably most familiar as the mathematical representation of Ising models, which are simplified, statistical models of the interaction of arrays of magnetic spins (Kindermann and Snell, 1980). MLNs thus combine (and generalize) FOL and MRFs to gain most of the advantages of both the logic and probabilistic modeling worlds. Among the advantages of MLN are the ability to handle arbitrary classes of variables, recursive statements, and the ability to construct multiple templates for MRFs. Using MLN can allow us to perform sophisticated probabilistic modeling while directly incorporating biological knowledge, particularly including partial knowledge, into the models, which is one of our major motivations for developing this general approach. We think that this capability is fundamentally important for the development of system level analysis and modeling of biological systems, including metabolic, regulatory, and genetic aspects of these systems. It is useful here to presage, or summarize, the conclusions of the later sections here by giving a compact overview of the key relationships that put the MLN and other methods into some context. We use the examples elaborated later in this article to illustrate this overview. Since the most widely used probabilistic graphical models in computational biology are various versions of BN, it is specifically useful to note the key differences between these and MLN. The major differences are these three: 1. The probability distributions that can be represented in MLN are those of MRF, which are more general than those of BN. 2. Relationships and probabilities can be assigned to classes of elements and/or events in MLN, whereas in BN probabilities are assigned to individual elements or events. 3. The representation of the logical relationships in MLN is much more compact than in BN (for those that can be expressed in BN), particularly for more complex models. 318 SAKHANENKO AND GALAS These differences will be illustrated and discussed in later sections. A fourth difference is that MLNs are much more computationally intensive to implement, which can be an important practical limitation. Now we turn to a specific example of model selection in genetic analysis. We have applied MLN by using a search method that allows us to solve various model selection problems with different biological knowledge constraints naturally embedded in them. In a previous article, we used MLN to select models for the influence of the genotype (specific sets of markers, indicating gene variants) on the phenotype (e.g., properties of the organism as expressed in measurements: size, color, shape, gene expression levels). We first used a single-marker model iteratively that focused on relationships between genetic markers and a given phenotype (Sakhanenko and Galas, 2010). Every marker associated with a gene selected in the model had a specific influence on the phenotype, even when considered alone. The model is asked to predict the phenotype given the genotype of a single organism, and the markers were iteratively selected to improve the prediction. Later, in this review, we discuss a natural extension of this application that uses a pair-wise gene model that explicitly represents relationships between a phenotype and pair-wise interacting genes. The models discussed here to illustrate the approach, while complex in implementation, are still rather simple genetic models, and the full power of MLNs will come with the natural extension to much more complex models. This is one of the major points